= Trine state ensemble
{title2=$p_j=1/3,\quad |t_j\rangle=(|0\rangle+e^{2\pi ij/3}|1\rangle)/\sqrt2$}
Three equally weighted pure <qubit> states with coplanar unit <Bloch vectors> separated by $120$ degrees. One example is $|t_j\rangle=(|0\rangle+\omega^j|1\rangle)/\sqrt2$, with $\omega=e^{2\pi i/3}$ and $j=0,1,2$. The roots-of-unity identity $1+\omega+\omega^2=0$ cancels the average <off-diagonal matrix entries>, so $\frac13\sum_j|t_j\rangle\langle t_j|=I/2$.
This is an ensemble of preparation states, whereas a <trine qubit POVM> consists of measurement effects $E_j=\frac23|t_j\rangle\langle t_j|$. The <POVM–ensemble duality for the maximally mixed state> relates them but does not identify their operational roles.
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